The Imbalanced Fermi Gas at Unitarity

Olga Goulko

In collaboration with Matthew WingateBased on Phys.Rev.A 82, 053621 (2010)

DAMTP, University of Cambridge

Jefferson Lab Theory Seminar, 2 May 2011

Cold quantum gases

• High tunability• interaction strength• temperature• dimensionality• . . .

• Many analogues• atomic gases• neutron stars• quark-gluon plasma• . . .

The unitary Fermi gas

Interacting system of two-component fermions:Low-energy interactions are characterised by the scattering length a

∞ −∞0

1a

BEC regimestrongly bound

(bosonic) moleculesof two fermions

UNITARITYstrongly

interactingfermions

BCS regimepairs of fermionsweakly bound in

momentum space

What is interesting about unitarity?

• System is dilute (range of potential � interparticle distance)and strongly interacting (interparticle distance � scatteringlength) at the same time

• No length scales associated with interactions ⇒ universalbehaviour

• Only relevant parameters: temperature and density

• High-temperature superfluidity

neutron star Tc = 106K Tc = 10−5TF

high-Tc superconductor Tc = 102K Tc = 10−3TF

atomic Fermi gas Tc = 10−7K Tc = 10−1TF

• Experimental data available

Methods to study unitarity

Strong interactions ⇒ No small parameter for perturbation theory

No exact theory for Fermi gas at unitarity!

What to do?

• Approximate schemes (e.g. mean-field theory) involveuncontrolled approximations

• Numerical Methods=⇒ Good results for critical temperature and other quantities

Our project: Calculating the critical temperature of the imbalancedunitary Fermi gas with the Determinant Diagrammatic MonteCarlo (DDMC) algorithm [Burovski et al. cond-mat/0605350]

The Fermi-Hubbard model

Simplest lattice model for two-particle scattering

• Non-relativistic fermions

• Contact interaction between spin up and spin down

• On-site attraction U < 0 tuned to describe unitarity

• Grand canonical ensemble

• Finite 3D simple cubic lattice, periodic boundary conditions

• Continuum limit can be taken by extrapolation to zero density

H =∑k,σ

(εk − µσ)c†kσckσ + U∑x

c†x↑cx↑c†x↓cx↓,

where εk = 1m

∑3j=1(1− cos kj) is the discrete FT of −∇

2

2m .

Finite temperature formalism

Grand canonical partition function in imaginary time interactionpicture: Z = Tre−βH :

Z = 1 + + +− − ± . . .

Sign problem!

The diagrams of each order can be written as the product of twomatrix determinants [Rubtsov et al. cond-mat/0411344]

Z =∑p,Sp

(−U)p detA↑(Sp) detA↓(Sp),

where Sp is the vertex configuration and the matrix entries are free(finite temperature) propagators

Order parameter of the phase transition

Anomalous correlations in the superfluid phase:

⇒ Introduce pair annihilation/creation operators P and P†:

P(x, τ) = cx↑(τ)cx↓(τ) and P†(x′, τ ′) = c†x′↑(τ′)c†x′↓(τ

′)

At the critical point the correlation function

G2(xτ ; x′τ ′) =⟨TτP(x, τ)P†(x′, τ ′)

⟩=

1

ZTrTτP(x, τ)P†(x′, τ ′)e−βH

is proportional to |x− x′|−(1+η) as |x− x′| → ∞(in 3 spatial dimensions, where η ≈ 0.038 for U(1) universalityclass)

Order parameter of the phase transition

⇒ the rescaled integrated correlation function

R(L,T ) = L1+ηG2(xτ ; x′τ ′)

becomes independent of lattice size at the critical point

Finite-size corrections:

R(L,T ) = (f0 + f1(T − Tc)L1/νξ + . . .)︸ ︷︷ ︸universal scaling function

(1 + cL−ω + . . .)︸ ︷︷ ︸finite-size scaling

• Critical exponents for the U(1) universality class:νξ ≈ 0.67 and ω ≈ 0.8

• Non-universal constants to be determined:Tc , f0, f1, c (to first order)

Order parameter of the phase transition

Crossing of R(L,T ) curves for 2 lattice sizes Li , Lj :

R(Li ,Tij) = R(Lj ,Tij)⇒ Tij − Tc = const. · g(Li , Lj)

with

g(Li , Lj) =(Lj/Li )

ω − 1

L1νξ

+ω

j

(1− (Li/Lj)

1νξ

)+ cL

1νξ

j

(1− (Li/Lj)

1νξ−ω)

︸ ︷︷ ︸neglect?

c can take values of O(1)⇒ perform non-linear fit to 4 parametersinstead

Order parameter of the phase transition

Example: fit of the rescaled integrated correlator R(L,T )

0.130.14

0.15T

10

12

14

16L

0.04

0.06

0.08

R

(data taken at 4 different temperatures and 4 different lattice sizes)

Diagrammatic Monte Carlo

Burovski et al. cond-mat/0605350:

• sampling via a Monte Carlo Markov chain process

• the configuration space is extended → worm vertices

• physical picture: at lowdensities multi-ladder diagramsdominate

• updates designed to favourprolonging existing vertexchains

The worm updates

Updates of the regular 4-point vertices: adding/removing a4-point vertex (changes the diagram order)

• Diagonal version: add or remove a random vertex

• Alternative using worm: move the P(x, τ) vertex to anotherposition and insert a 4-point vertex at its old position.⇒ choose new coordinates of P very close to its initialcoordinates⇒ the removal update always attempts to remove the nearestneighbour of P

AutocorrelationsThe original worm algorithm achieved high acceptance ratios, butat the cost of strongly autocorrelated results:

40

60

80

100

120

140

160

0 20000 40000 60000 80000 100000

40

60

80

100

120

140

160

0 20000 40000 60000 80000 100000

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 10000 20000 30000 40000 50000 60000 70000

Rela

tive E

rror

Block Size

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 10000 20000 30000 40000 50000 60000 70000

Rela

tive E

rror

Block Size

Worm updates Diagonal updates

Alternative updates

Alternative set of updates: both weak autocorrelations and highacceptance rates [Goulko and Wingate, arXiv:0910.3909].

• Choose a random 4-point vertex from the configuration (willact as a worm for this step).

• Addition: add another 4-point vertex on the same lattice siteand in some time interval around the worm.

• Removal: remove the nearest neighbour of the worm vertex

This setup still prolongs existing vertex chains, but autocorrelationsare reduced since the worm changes with every update.

Alternative updates

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0 20 40 60 80 100 120 140

Rela

tive E

rror

Block Size

Comparison between diagonal setup (red circles) and alternativeworm setup (blue squares) at low filling factor

The balanced Fermi gas

An interacting system with equal number of spin up and spin downfermions (µ↑ = µ↓)

The imbalanced Fermi gas

Interactions are suppressed in presence of an imbalance (µ↑ 6= µ↓)

The imbalanced Fermi gas

Thermal probability distribution:

ρ(Sp) =1

Z(−U)p detA↑(Sp) detA↓(Sp)

Sign problem: µ↑ 6= µ↓ ⇒ detA↑ detA↓ 6= | detA|2

Sign quenched method: write ρ(Sp) = |ρ(Sp)|sign(Sp) and use|ρ(Sp)| as the new probability distribution

〈X 〉ρ =

∑X (Sp)ρ(Sp)∑

ρ(Sp)=

∑X (Sp)|ρ(Sp)|sign(Sp)∑ |ρ(Sp)|sign(Sp)

=〈X sign〉|ρ|〈sign〉|ρ|

Problems if 〈sign〉 ≈ 0

But for the unitary Fermi gas 〈sign〉|ρ| ≈ 1 for a wide range of ∆µ

The imbalanced Fermi gas

Schematic plot of the sign near the critical point:

0.00 0.05 0.10 0.15 0.20

DΜ

¶F

0.2

0.4

0.6

0.8

1.0

<sign>

Results

Relationship between ∆µ/εF = |µ↑ − µ↓|/εF and δν/ν =|ν↑−ν↓|ν↑+ν↓

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.05 0.1 0.15 0.2 0.25

∆ν/ν

∆µ/εF

Results: the critical temperature

The critical temperature using only balanced data:

0.05

0.1

0.15

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Tc/ε

F

ν1/3

Tc(ν = 0) = 0.173(6)εF

ν → 0 corresponds to the continuum limit

Surface fits for the imbalanced gas

Surface fit of a physical observable X as a function of filling factorν1/3 and imbalance h = ∆µ/εF :

• At fixed imbalance X is a linear function of ν1/3, with slopeα(X )(h).

• X (h) and α(X )(h) viewed as functions h can be Taylorexpanded.

• Due to symmetry in h all odd powers in the expansion of X (h)and α(X )(h) have to vanish.

Hence the fitted function takes the form

X (ν, h) = X (h) + α(X )(h)ν1/3

We will expand the functions X (h) and α(X )(h) to 2nd order in h.

Results: the critical temperature

Surface fit of the critical temperature versus ν1/3 and h:

0.0 0.2 0.4 0.6

Υ1�3

0.0

0.1

0.2

DΜ�¶F

0.05

0.10

0.15

Tc�¶F

Data is consistent with Tc(ν = 0) = 0.171(5)εF , independent of h.

Results: the critical temperature

Lower bounds for the deviation of Tc from the balanced limit:

0.00 0.05 0.10 0.15 0.20 0.25

DΜ�¶F

0.14

0.15

0.16

0.17

Tc�¶F

lower bound: Tc(h)− Tc(0) > −0.5εFh2,

with additional assumption: Tc(h)− Tc(0) > −0.04εFh2

Results: the critical temperature

Comparison with other numerical studies and experiment:

• Crossings• Burovski, Prokof’ev, Svistunov, Troyer (DDMC) 0.152(7)• Burovski, Kozik, Prokof’ev, Svistunov, Troyer 0.152(9)• Bulgac, Drut, Magierski 0.15(1)

• Full fit• Abe, Seki 0.189(12)• Goulko, Wingate (DDMC) 0.171(5)

• Experiment• Nascimbene, Navon, Jiang, Chevy, Salomon 0.157(15)• Horikoshi, Nakajima, Ueda, Mukaiyama 0.17(1)

Results: the chemical potential

The average chem. pot. projected onto the (ν1/3 − µ) plane:

0.2

0.25

0.3

0.35

0.4

0.45

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

µ/ε

F

ν1/3

µ(ν = 0) = 0.429(7)εF

ν → 0 corresponds to the continuum limit

Results: the energy per particle

The energy per particle using only balanced data:

0.25

0.3

0.35

0.4

0.45

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

E/E

FG

ν1/3

E(ν = 0) = 0.46(2)EFG

ν → 0 corresponds to the continuum limit; EFG = (3/5)NεF

Results: the energy per particleSurface fit of the energy per particle versus ν1/3 and h:

0.0 0.2 0.4 0.6 0.8

Υ1�3

0.00

0.050.10

0.150.20

DΜ�¶F

0.3

0.4

0.5

0.6

E�EFG

Results: the contact density

The contact can be interpreted as a measure for the local pairdensity [Braaten, arXiv:1008.2922].

Definition contact [Werner and Castin, arXiv:1001.0774]:

C = m2g0Eint,

where g0 is the physical coupling constant.

The contact density is C = C/V and has units ε2F .

This was the first numerical calculation of the contact density atfinite temperature [Goulko and Wingate, arXiv:1011.0312]

Results: the contact density

The contact density using only balanced data:

0.08

0.085

0.09

0.095

0.1

0.105

0.11

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

C/ε

F2

ν1/3

C(ν = 0) = 0.1102(11)εF2

ν → 0 corresponds to the continuum limit

Results: the contact densitySurface fit of the contact density versus ν1/3 and h:

0.0 0.2 0.4 0.6 0.8

Υ1�3

0.0

0.1

0.2

DΜ�¶F

0.09

0.10

0.11

C�¶F2

Outlook: temperatures beyond Tc

Problem: fixing a physical isotherm for T 6= Tc

Setting the scale: set lattice spacing b to be independent oftemperature ⇒ b = b(µ)

⇒ ν(µ,T )

ν(µ,Tc)=

n(T )

n(Tc)

(b(µ,T )

b(µ,Tc)

)3

=n(T )

n(Tc)

If the fix the lattice temperature ratio T (µ)/Tc(µ) we will movealong an isotherm

Outlook: temperatures beyond Tc

Works for T/Tc ≤ 4 for sufficiently small µ:

0

2

4

6

8

10

0 2 4 6 8 10

ν(µ

,T)/ν(µ

,Tc)

T/Tc

µ=0.4µ=0.5µ=0.7

Outlook: temperatures beyond Tc

Preliminary results: temperature dependence of the chemicalpotential

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

µ/ε

F

ν1/3

T/Tc=3

T/Tc=2

T/Tc=1.5

T/Tc=1

Outlook: temperatures beyond Tc

Preliminary results: temperature dependence of the contact

0.075

0.1

0.125

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

C/ε

F

2

ν1/3

T/Tc=3

T/Tc=2

T/Tc=1.5

T/Tc=1

Conclusions

• Lattice Field Theory is a useful tool for studying stronglyinteracting systems in condensed matter physics

• The DDMC algorithm can be applied to study the phasetransition of the unitary Fermi gas

• Imbalanced case with the sign quenched method

• Results for Tc/εF , µ/εF , E/EFG and C/ε2F for equal andunequal number of fermions in the two spin components

• Temperatures beyond Tc accessible

Thank you!